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Table 1 Estimates of ω assuming the standard expression from thin-beam theory for the third in-plane flexural frequency with clamped-clamped boundary conditions

From: Quantum simulation of the Anderson Hamiltonian with an array of coupled nanoresonators: delocalization and thermalization effects

ω /2 π (GHz) Δ V (V) λ /2 π (MHz) J /2 π (MHz)
2.5 10 1.2 0.7
2.5 20 2.3 2.7
3.5 10 1.2 0.6
3.5 20 2.5 2.3
  1. The nanoresonator is assumed to be aluminum, with the following dimensions: length L = 0.70 μm (0.6 μm) for the 2.5 GHz (3.5 GHz) mode; width w = 45 nm; thickness t = 50 nm; The transmon-NR coupling λ was calculated from circuit theory, assuming the systems to be in resonance, and is given by \(\lambda= \omega\sqrt{\frac{dC}{dx}\frac{C \Delta V ^{2}}{m \omega ^{2} d C_{Q}}}\). Here C = 20aF, dC/dx = 6 × 10−11 F/m are the coupling capacitance and its first derivative respectively, which are estimated from finite element simulations assuming a gap of d = 20 nm. \(C_{Q}=50 fF\) is the transmon shunt capacitance, which would yield a charging energy of \(E_{C}/h = 400\) MHz. Finally, m = 0.52. ρwtL is the effective mass of the third mode, where ρ is the density of aluminum. The same parameter values are assumed for the calculation of J.